src/HOL/HOL.ML
author nipkow
Mon Oct 21 09:50:50 1996 +0200 (1996-10-21)
changeset 2115 9709f9188549
parent 2031 03a843f0f447
child 2442 6663e0d210b0
permissions -rw-r--r--
Added trans_tac (see Provers/nat_transitive.ML)
     1 (*  Title:      HOL/HOL.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1991  University of Cambridge
     5 
     6 For HOL.thy
     7 Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68 
     8 *)
     9 
    10 open HOL;
    11 
    12 
    13 (** Equality **)
    14 section "=";
    15 
    16 qed_goal "sym" HOL.thy "s=t ==> t=s"
    17  (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
    18 
    19 (*calling "standard" reduces maxidx to 0*)
    20 bind_thm ("ssubst", (sym RS subst));
    21 
    22 qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
    23  (fn prems =>
    24         [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
    25 
    26 (*Useful with eresolve_tac for proving equalties from known equalities.
    27         a = b
    28         |   |
    29         c = d   *)
    30 qed_goal "box_equals" HOL.thy
    31     "[| a=b;  a=c;  b=d |] ==> c=d"  
    32  (fn prems=>
    33   [ (rtac trans 1),
    34     (rtac trans 1),
    35     (rtac sym 1),
    36     (REPEAT (resolve_tac prems 1)) ]);
    37 
    38 
    39 (** Congruence rules for meta-application **)
    40 section "Congruence";
    41 
    42 (*similar to AP_THM in Gordon's HOL*)
    43 qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)"
    44   (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
    45 
    46 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
    47 qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)"
    48  (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
    49 
    50 qed_goal "cong" HOL.thy
    51    "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
    52  (fn [prem1,prem2] =>
    53    [rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);
    54 
    55 
    56 (** Equality of booleans -- iff **)
    57 section "iff";
    58 
    59 qed_goal "iffI" HOL.thy
    60    "[| P ==> Q;  Q ==> P |] ==> P=Q"
    61  (fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]);
    62 
    63 qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P"
    64  (fn prems =>
    65         [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
    66 
    67 val iffD1 = sym RS iffD2;
    68 
    69 qed_goal "iffE" HOL.thy
    70     "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
    71  (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
    72 
    73 
    74 (** True **)
    75 section "True";
    76 
    77 qed_goalw "TrueI" HOL.thy [True_def] "True"
    78   (fn _ => [rtac refl 1]);
    79 
    80 qed_goal "eqTrueI " HOL.thy "P ==> P=True" 
    81  (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
    82 
    83 qed_goal "eqTrueE" HOL.thy "P=True ==> P" 
    84  (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
    85 
    86 
    87 (** Universal quantifier **)
    88 section "!";
    89 
    90 qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
    91  (fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]);
    92 
    93 qed_goalw "spec" HOL.thy [All_def] "! x::'a.P(x) ==> P(x)"
    94  (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
    95 
    96 qed_goal "allE" HOL.thy "[| !x.P(x);  P(x) ==> R |] ==> R"
    97  (fn major::prems=>
    98   [ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]);
    99 
   100 qed_goal "all_dupE" HOL.thy 
   101     "[| ! x.P(x);  [| P(x); ! x.P(x) |] ==> R |] ==> R"
   102  (fn prems =>
   103   [ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]);
   104 
   105 
   106 (** False ** Depends upon spec; it is impossible to do propositional logic
   107              before quantifiers! **)
   108 section "False";
   109 
   110 qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
   111  (fn [major] => [rtac (major RS spec) 1]);
   112 
   113 qed_goal "False_neq_True" HOL.thy "False=True ==> P"
   114  (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
   115 
   116 
   117 (** Negation **)
   118 section "~";
   119 
   120 qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
   121  (fn prems=> [rtac impI 1, eresolve_tac prems 1]);
   122 
   123 qed_goalw "notE" HOL.thy [not_def] "[| ~P;  P |] ==> R"
   124  (fn prems => [rtac (prems MRS mp RS FalseE) 1]);
   125 
   126 qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
   127  (fn _ => [REPEAT (ares_tac [notE] 1)]);
   128 
   129 
   130 (** Implication **)
   131 section "-->";
   132 
   133 qed_goal "impE" HOL.thy "[| P-->Q;  P;  Q ==> R |] ==> R"
   134  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
   135 
   136 (* Reduces Q to P-->Q, allowing substitution in P. *)
   137 qed_goal "rev_mp" HOL.thy "[| P;  P --> Q |] ==> Q"
   138  (fn prems=>  [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
   139 
   140 qed_goal "contrapos" HOL.thy "[| ~Q;  P==>Q |] ==> ~P"
   141  (fn [major,minor]=> 
   142   [ (rtac (major RS notE RS notI) 1), 
   143     (etac minor 1) ]);
   144 
   145 qed_goal "rev_contrapos" HOL.thy "[| P==>Q; ~Q |] ==> ~P"
   146  (fn [major,minor]=> 
   147   [ (rtac (minor RS contrapos) 1), (etac major 1) ]);
   148 
   149 (* ~(?t = ?s) ==> ~(?s = ?t) *)
   150 bind_thm("not_sym", sym COMP rev_contrapos);
   151 
   152 
   153 (** Existential quantifier **)
   154 section "?";
   155 
   156 qed_goalw "exI" HOL.thy [Ex_def] "P(x) ==> ? x::'a.P(x)"
   157  (fn prems => [rtac selectI 1, resolve_tac prems 1]);
   158 
   159 qed_goalw "exE" HOL.thy [Ex_def]
   160   "[| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q"
   161   (fn prems => [REPEAT(resolve_tac prems 1)]);
   162 
   163 
   164 (** Conjunction **)
   165 section "&";
   166 
   167 qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
   168  (fn prems =>
   169   [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
   170 
   171 qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
   172  (fn prems =>
   173    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
   174 
   175 qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
   176  (fn prems =>
   177    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
   178 
   179 qed_goal "conjE" HOL.thy "[| P&Q;  [| P; Q |] ==> R |] ==> R"
   180  (fn prems =>
   181          [cut_facts_tac prems 1, resolve_tac prems 1,
   182           etac conjunct1 1, etac conjunct2 1]);
   183 
   184 
   185 (** Disjunction *)
   186 section "|";
   187 
   188 qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
   189  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
   190 
   191 qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
   192  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
   193 
   194 qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
   195  (fn [a1,a2,a3] =>
   196         [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
   197          rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
   198 
   199 
   200 (** CCONTR -- classical logic **)
   201 section "classical logic";
   202 
   203 qed_goalw "classical" HOL.thy [not_def]  "(~P ==> P) ==> P"
   204  (fn [prem] =>
   205    [rtac (True_or_False RS (disjE RS eqTrueE)) 1,  assume_tac 1,
   206     rtac (impI RS prem RS eqTrueI) 1,
   207     etac subst 1,  assume_tac 1]);
   208 
   209 val ccontr = FalseE RS classical;
   210 
   211 (*Double negation law*)
   212 qed_goal "notnotD" HOL.thy "~~P ==> P"
   213  (fn [major]=>
   214   [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
   215 
   216 qed_goal "contrapos2" HOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [
   217         rtac classical 1,
   218         dtac p2 1,
   219         etac notE 1,
   220         rtac p1 1]);
   221 
   222 qed_goal "swap2" HOL.thy "[| P;  Q ==> ~ P |] ==> ~ Q" (fn [p1,p2] => [
   223         rtac notI 1,
   224         dtac p2 1,
   225         etac notE 1,
   226         rtac p1 1]);
   227 
   228 (** Unique existence **)
   229 section "?!";
   230 
   231 qed_goalw "ex1I" HOL.thy [Ex1_def]
   232             "[| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
   233  (fn prems =>
   234   [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
   235 
   236 qed_goalw "ex1E" HOL.thy [Ex1_def]
   237     "[| ?! x.P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R"
   238  (fn major::prems =>
   239   [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
   240 
   241 
   242 (** Select: Hilbert's Epsilon-operator **)
   243 section "@";
   244 
   245 (*Easier to apply than selectI: conclusion has only one occurrence of P*)
   246 qed_goal "selectI2" HOL.thy
   247     "[| P(a);  !!x. P(x) ==> Q(x) |] ==> Q(@x.P(x))"
   248  (fn prems => [ resolve_tac prems 1, 
   249                 rtac selectI 1, 
   250                 resolve_tac prems 1 ]);
   251 
   252 qed_goal "select_equality" HOL.thy
   253     "[| P(a);  !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a"
   254  (fn prems => [ rtac selectI2 1, 
   255                 REPEAT (ares_tac prems 1) ]);
   256 
   257 qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) =  (? x. P x)" (fn prems => [
   258         rtac iffI 1,
   259         etac exI 1,
   260         etac exE 1,
   261         etac selectI 1]);
   262 
   263 
   264 (** Classical intro rules for disjunction and existential quantifiers *)
   265 section "classical intro rules";
   266 
   267 qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
   268  (fn prems=>
   269   [ (rtac classical 1),
   270     (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
   271     (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
   272 
   273 qed_goal "excluded_middle" HOL.thy "~P | P"
   274  (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
   275 
   276 (*For disjunctive case analysis*)
   277 fun excluded_middle_tac sP =
   278     res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
   279 
   280 (*Classical implies (-->) elimination. *)
   281 qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R" 
   282  (fn major::prems=>
   283   [ rtac (excluded_middle RS disjE) 1,
   284     REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
   285 
   286 (*Classical <-> elimination. *)
   287 qed_goal "iffCE" HOL.thy
   288     "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
   289  (fn major::prems =>
   290   [ (rtac (major RS iffE) 1),
   291     (REPEAT (DEPTH_SOLVE_1 
   292         (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
   293 
   294 qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x.P(x)"
   295  (fn prems=>
   296   [ (rtac ccontr 1),
   297     (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ]);
   298 
   299 
   300 (* case distinction *)
   301 
   302 qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
   303   (fn [p1,p2] => [cut_facts_tac [excluded_middle] 1, etac disjE 1,
   304                   etac p2 1, etac p1 1]);
   305 
   306 fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
   307 
   308 
   309 (** Standard abbreviations **)
   310 
   311 fun stac th = CHANGED o rtac (th RS ssubst);
   312 fun strip_tac i = REPEAT(resolve_tac [impI,allI] i); 
   313 
   314 (** strip proved goal while preserving !-bound var names **)
   315 
   316 local
   317 
   318 (* Use XXX to avoid forall_intr failing because of duplicate variable name *)
   319 val myspec = read_instantiate [("P","?XXX")] spec;
   320 val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
   321 val cvx = cterm_of (#sign(rep_thm myspec)) vx;
   322 val aspec = forall_intr cvx myspec;
   323 
   324 in
   325 
   326 fun RSspec th =
   327   (case concl_of th of
   328      _ $ (Const("All",_) $ Abs(a,_,_)) =>
   329          let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
   330          in th RS forall_elim ca aspec end
   331   | _ => raise THM("RSspec",0,[th]));
   332 
   333 fun RSmp th =
   334   (case concl_of th of
   335      _ $ (Const("op -->",_)$_$_) => th RS mp
   336   | _ => raise THM("RSmp",0,[th]));
   337 
   338 fun normalize_thm funs =
   339 let fun trans [] th = th
   340       | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
   341 in trans funs end;
   342 
   343 fun qed_spec_mp name =
   344   let val thm = normalize_thm [RSspec,RSmp] (result())
   345   in bind_thm(name, thm) end;
   346 
   347 end;
   348 
   349 
   350 (*Thus, assignments to the references claset and simpset are recorded
   351   with theory "HOL".  These files cannot be loaded directly in ROOT.ML.*)
   352 use "hologic.ML";
   353 use "cladata.ML";
   354 use "simpdata.ML";